If the lines `x+y+3=0 and 2x-y-9=0` lie along diameters of a circle of circumference `8pi` then the equation of the circle is :

Options

A. `x^(2)+y^(2)-4x+10y+13=0`
B. `x^(2)+y^(2)+4x-10y+13=0`
C. `x^(2)+y^(2)-4x-10y+13=0`
D. `x^(2)+y^(2)+4x+10y+13=0`

Explanation

To find the equation of the circle given the lines \(x + y + 3 = 0\) and \(2x - y - 9 = 0\) that lie along the diameters of the circle with a circumference of \(8\pi\), we can follow these steps:

### Step 1: Find the intersection point of the lines
To find the center of the circle, we need to determine the intersection point of the two lines.

1. The first line is given by:
\[ x + y + 3 = 0 \quad \text{(1)} \]
Rearranging gives:
\[ y = -x - 3 \]

2. The second line is given by:
\[ 2x - y - 9 = 0 \quad \text{(2)} \]
Rearranging gives:
\[ y = 2x - 9 \]

Now, we can set the two expressions for \(y\) equal to each other:
\[ -x - 3 = 2x - 9 \]

### Step 2: Solve for \(x\)
Combining like terms:
\[ -x - 2x = -9 + 3 \]
\[ -3x = -6 \]
\[ x = 2 \]

### Step 3: Solve for \(y\)
Substituting \(x = 2\) back into one of the equations to find \(y\):
Using equation (1):
\[ y = -2 - 3 = -5 \]

Thus, the center of the circle is at the point \((2, -5)\).

### Step 4: Calculate the radius
The circumference of the circle is given as \(8\pi\). The formula for the circumference \(C\) of a circle is:
\[ C = 2\pi r \]
Setting this equal to \(8\pi\):
\[ 2\pi r = 8\pi \]
Dividing both sides by \(2\pi\):
\[ r = 4 \]

### Step 5: Write the equation of the circle
The standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \(h = 2\), \(k = -5\), and \(r = 4\):
\[ (x - 2)^2 + (y + 5)^2 = 4^2 \]
\[ (x - 2)^2 + (y + 5)^2 = 16 \]

### Step 6: Expand the equation
Expanding the equation:
\[ (x^2 - 4x + 4) + (y^2 + 10y + 25) = 16 \]
Combining like terms:
\[ x^2 + y^2 - 4x + 10y + 29 - 16 = 0 \]
\[ x^2 + y^2 - 4x + 10y + 13 = 0 \]

Thus, the equation of the circle is:
\[ x^2 + y^2 - 4x + 10y + 13 = 0 \]

### Final Answer
The equation of the circle is:
\[ x^2 + y^2 - 4x + 10y + 13 = 0 \]